Geometric Characterization of Property R

الأصل · EN

Consider pairs of the form (G, N), with G a group and N G, as objects of a category. A morphism (G₁, N₁) (G₂, N₂) will be a group homomorphism f: G₁ G₂ such that f(N₁) ⊂ N₂. We introduce a functor Q:, which provides a geometric definition of Property R, since it is most naturally visualized by means of a directed graph. We compute these graphs for a number of finite groups of small order, and prove a general characterization of the graphs which occur in this way.